Estimator Variance

As mentioned in §6.12, the pwelch function in Matlab
and Octave offer ``confidence intervals'' for an estimated power
spectral density (PSD). A confidence interval encloses the
true value with probability
(the confidence level). For
example, if
, then the confidence level is
.

This section gives a first discussion of ``estimator variance,''
particularly the variance of sample means and sample
variances for stationary stochastic processes.

Here we have defined the sample mean at time
as the average of the
successive samples up to time
--a ``running average''. The
true mean is assumed to be the average over any infinite number of
samples such as

(C.30)

or

(C.31)

Now assume
, and let
denote the
variance of the process
, i.e.,

Var

(C.32)

Then the variance of our sample-mean estimator
can be calculated as follows:

where we used the fact that the time-averaging operator
is
linear, and
denotes the unbiased autocorrelation of
.
If
is white noise, then
, and we obtain

We have derived that the variance of the
-sample running average of
a white-noise sequence
is given by
, where
denotes the variance of
. We found that the
variance is inversely proportional to the number of samples used to
form the estimate. This is how averaging reduces variance in general:
When averaging
independent (or merely uncorrelated) random
variables, the variance of the average is proportional to the variance
of each individual random variable divided by
.

where the mean is assumed to be
, and
denotes the unbiased sample autocorrelation of
based on the
samples leading up to and including time
. Since
is unbiased,
.
The variance of this estimator is then given by

where

The autocorrelation of
need not be simply related to that of
. However, when
is assumed to be Gaussianwhite
noise, simple relations do exist. For example, when
,

(C.34)

by the independence of
and
, and when
,
the fourth moment is given by
.
More generally, we can simply label the
th moment of
as
, where
corresponds to the mean,
corresponds to the variance (when the mean is zero), etc.

so that the variance of our estimator for the variance of Gaussian
white noise is

Var

(C.36)

Again we see that the variance of the estimator declines as
.

The same basic analysis as above can be used to estimate the variance
of the sample autocorrelation estimates for each lag, and/or the
variance of the power spectral density estimate at each frequency.